MATH 308 Lecture 25

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Lecture Notes

Convolution Function

Defined

$(f*g)(t)=\int _{0}^{t}f(t-v)\,g(v)\,\mathrm {d} v=\int _{0}^{t}f(u)\,g(t-u)\,\mathrm {d} u$

• It is associative, so $f*g=g*f$
• It is distributive, so $(f*(g(t)+a\,h(t)))(t)=(f*g)(t)+a(f*h)(t)$
• $f*0(t)=0$

Example

$f(t)=\mathrm {e} ^{t}$, and $g(t)=\sin {(3\,t)}$:

$(f*g)(t)=\int _{0}^{t}\mathrm {e} ^{t-v}\,\sin {3v}\,\mathrm {d} v=\int _{0}^{t}\mathrm {e} ^{t}\,\sin {(3(t-v))}\,\mathrm {d} v$

$f(t)=t^{2}$, and $g(t)=1$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f * g)(t) = \int_0^t v^2 \cdot 1 \, \mathrm{d}v = \frac{t^3}{3}}

Theorem 6.6.1

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L} \left\{ f * g \right\} = \mathcal{L} \left\{ f \right\} \, \mathcal{L} \left\{ g \right\}}

So finding the inverse Laplace transform of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{(s-7)^9 \, (s^2+4)}} is equivalent to finding the convolution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{(t^8 \, \mathrm{e}^{7t}) * (\sin{2t})}{8! \cdot 2}}

Exercises

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}^{-1} \left\{ \frac{1}{(s-1)(s^2+9)} \right\} = \mathrm{e}^{t} * \frac{\sin{3t}}{3}}
2. ${\mathcal {L}}\left\{g(t)\right\}=G(s)$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(s) = G(s) \, \frac{1}{s^2+1} = \mathcal{L} \left\{ g(t) * \sin{t} \right\}}

Exercise 3

Find Laplace transforms of

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t) = \int_0^t (t-v)^3 \, \sin{2v} \, \mathrm{d}v = t^3 * \sin{2t}} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L} \left\{ f(t) \right\} = \frac{6}{s^4} \, \frac{2}{s^2+4} = \frac{12}{s^4(s^2+4)}}
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(t) = \int_0^t \mathrm{e}^{v-t} \, \cos{v} \, \mathrm{d}v = \mathrm{e}^{-t} * \cos{t}} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L} \left\{ g(t) \right\} = \frac{s}{s^2+1} \, \frac{1}{s+1} = \frac{s}{(s^2+1)(s+1)}}

Solving Differential Equations

Find the solution to $y''+\omega ^{2}\,y=g(t)$ when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(0) = 0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'(0) = 1} .

Take Laplace transform of both sides:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s^2 \mathcal{L} \left\{ y \right\} - y'(0) - s \, y(0) + \omega^2 \, \mathcal{L} \left\{ y \right\} = \mathcal{L} \left\{ g(t) \right\}}

Solve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L} \left\{ y \right\}} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L} \left\{ y \right\} = \frac{\mathcal{L} \left\{ g(t) \right\}}{s^2+\omega^2} + \frac{1}{s^2 + \omega^2}}

So the solution is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = g(t) * \frac{\sin{\omega\,t}}{\omega} + \frac{\sin{\omega \, t}}{\omega}}

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Find the solution to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'' + 3y' + 2y = \cos{a\,t}} when $y(0)=1$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'(0) = 0}

We can solve this using two methods: undetermined coefficients and variation of parameters. Let's try to use Laplace transforms:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s^2 \, \mathcal{L} \left\{ y \right\} - y'(0) - s \, y(0) + 3 \mathcal{L} \left\{ y \right\} - 3y(0) + 2 \mathcal{L} \left\{ y \right\} = \frac{s}{s^2 + a^2}}

Solve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L} \left\{ y \right\}} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathcal{L} \left\{ y \right\} &= \frac{s}{(s^2+a^2)(s^2+3s+2)} + \frac{s+3}{s^2+3s+2} \\ &= \left( \frac{s}{s^2+a} \right) \, \left( \frac{1}{s^2+3s+2} \right) + \frac{s+3}{(s+2)(s+1)} \\ &= \mathcal{L} \left\{ \cos{a\,t} \right\} \, \mathcal{L} \left\{ \mathrm{e}^{-2t} \right\} \, \mathcal{L} \left\{ \mathrm{e}^{-t} \right\} + \frac{s+3}{(s+2)(s+1)} \\ &= \mathcal{L} \left\{ \cos{a\,t} \right\} \, \mathcal{L} \left\{ \mathrm{e}^{-2t} \right\} \, \mathcal{L} \left\{ \mathrm{e}^{-t} \right\} - \frac{1}{s+2} + 2 \, \frac{1}{s+1} \end{align}}

And then take the inverse Laplace transform:

(out of time)